Necessary and sufficient conditions for pseudo Riemannian manifold to be time orientable

Question

It is well known that a smooth manifold $M$ is orientable if the first Stiefel-Whitney class of the tangent bundle vanishes. In particular, this implies that if $M$ is equipped with a pseudo-Riemannian metric of signature $(t,s)$, then the frame bundle $O(M)$ admits a reduction to a principal $SO(t,s)$ bundle under the embedding $SO(t,s)\hookrightarrow O(t,s)$.

Now, let $SO^+(t,s)$ denote the Lie subgroup of $SO(t,s)$ consisting of the linear isometries of $\mathbb{R}^{t,s}$ which preserve the orientation of any maximally negative definite subspace of $\mathbb{R}^{t,s}$. The pseudo Riemannian manifold is then orientable and time orientable if the frame bundle $O(M)$ admits a reduction to a principal $SO^+(t,s)$ bundle under the embedding $SO^+(t,s)\hookrightarrow O(t,s)$. Is there a characteristic class related to this result?

I am asking because any orientable and time orientable pseudo Riemmanian manifolds admits a $\text{Spin}^+(t,s)$ structure if and only if the second Stiefel-Whitney class of $TM$ vanishes. If instead $M$ is just orientable, then I imagine that $M$ would admit a $\text{Spin}(t,s)$ structure if and only if the second Stiefel-Whitney class vanishes. However, I am unsure of when general orientable pseudo Riemannian manifolds which are also spin admit a $\text{Spin}^+(t,s)$ structure, i.e. when can we conclude that a pseudo Riemannian manifold that is spin, is also $\text{spin}^+$?

In the Lorentzian case, then we get a time orientation for free, as a necessary and sufficient condition for the existence of a Lorentz metric is a nowhere vanishing vector field, which would easily supply us with a time orientation.

I would guess that if there exists a vector bundle isomorphism: $$TM\cong E^t\oplus E^s$$ where the vector bundle $E^t$ has vanishing first Stiefel-Whitney class, then $TM$ is time orientable, but I am honestly unsure. Any source, or collection of sources that systematically treats existence of spin structures, and time orientations in the pseudo Riemannian case would also be greatly appreciated.

Answer 1

For the first question I believe the answer is yes. Almost certainly it's in the literature but I do not know this literature very well.

The idea is as you suggested. Maximal-rank timelike (or spacelike) vector subspaces form a convex space. So you paste together local decompositions using a partition of unity to decompose the tangent bundle of your Lorentz manifold into an orthogonal direct-sum of a maximal timelike sub-bundle and a maximal space-like subbundle.

You then have characteristic classes for the sub-bundles, and $w_1$ of your timelike sub-bundle is the obstruction you are looking for.